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\begin{center}
\vskip 1cm{\LARGE\bf A Generalization of the Catalan Numbers}
\vskip 1cm
\large
Reza Kahkeshani\\
Department of Pure Mathematics\\
Faculty of Mathematical Sciences\\
University of Kashan\\
Kashan\\
Iran\\
\href{mailto:kahkeshanireza@kashanu.ac.ir}{\tt kahkeshanireza@kashanu.ac.ir}\\
\ \\
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\begin{abstract}
In this paper, we generalize the Catalan number $C_n$ to the $(m,n)$th Catalan
number $C(m,n)$ using a combinatorial description, as follows:
the number of paths
in $\mathbb{R}^m$ from the origin to the point
$\bigl( \underbrace{n,\ldots,n}_{m-1},(m-1)n \bigr)$ with $m$ kinds of moves
such that the path never rises above the hyperplane
$x_m=x_1+\cdots+x_{m-1}$.
\end{abstract}
\begin{section}{Introduction}
Catalan numbers (\seqnum{A000108}) are a very prominent sequence of numbers that arises in
a wide varity of combinatorial situations \cite{B,G}.
Stanley \cite{S1} gave a list of 66 different combinatorial descriptions of
Catalan numbers and he added to the list some more \cite{S2}.
Some of the specific instances are
\begin{itemize}
\item The number of
movements in xy-plane from $(0,0)$ to $(n,n)$ with two kinds of moves
$$R:(x,y)\rightarrow(x+1,y),\quad U:(x,y)\rightarrow(x,y+1),$$
such that the path never rises above the line $y=x$.
\item Triangulations of a convex $(n+2)$-gon into $n$ triangles
by $n-1$ diagonals that do not intersect in their interiors.
\item Binary parenthesizations of a string of $n+1$ letters.
\item Binary trees with $n$ vertices.
\end{itemize}
The solution to these problems is the $n$th Catalan number
$$C_n=\frac{1}{n+1} {2n \choose n},$$
and the sequence $C_0,C_1,C_2,\ldots,C_n,\ldots$ is called the Catalan sequence.
There have been many attempts to generalize the Catalan numbers.
Probably the most important generalization consists of the $k$-ary numbers or
$k$-Catalan numbers, defined by
$$C_n^k=\frac{1}{kn+1} {kn+1 \choose n}=\frac{1}{(k-1)n+1} {kn \choose n}=%
\frac{1}{n} {kn \choose n-1},$$
where $k,n\in\mathbb{N}$. Clearly, $C_n^2=C_n$.
The $k$-good paths (below the line $y=kx$) from $(0,-1)$ to
$\bigl( n,(k-1)n-1 \bigr)$, staircase tilings and $k$-ary trees are structures
known to be enumerated by $k$-ary numbers \cite{HLM,HP,P,S1}.
Moreover, Kim \cite[Thm. 2]{K} showed that $C_n^k$ is the number of partitions
of $n(k-1)+2$ polygon by $(k+1)$-gon where all vertices of all $(k+1)$-gon
lie on the vertices of $n(k-1)+2$ polygon.
Gould \cite{Go} developed a generalization that has both the Catalan numbers and
the $k$-Catalan numbers as special cases, defined as
$$A_n(a,b)=\frac{a}{a+bn}{a+bn \choose n},$$
and showed the following convolution formula for these sequences:
$$\sum_{k=0}^{n}A_k(a,b)A_{n-k}(c,b)=A_{n}(a+c,b).$$
These numbers are also known as the Rothe numbers \cite{R} and Rothe-Hagen
coefficients of the first type \cite{Go1}.
Clearly, $A_n(1,2)=C_n$ and $A_n(1,k)=C_n^k$.
We know that one of the interpretations of the Catalan numbers is the movements
in $\mathbb{R}^2$ with two kinds of moves such that the path never rises above
the line $y=x$.
In this paper, a new generalization of the Catalan numbers using this
interpretation is introduced.
Consider $m$ kinds of moves in $\mathbb{R}^m$ such that they
are one unit parallel to the positive axes.
We show that the number of paths from the origin to the point
$\bigl( \underbrace{n,\ldots,n}_{m-1},(m-1)n \bigr)$ using these moves
such that the path never rises above the hyperplane
$x_m=x_1+\cdots+x_{m-1}$ is
$$\frac{1}{n(m-1)+1} {2n(m-1) \choose \underbrace{n,\ldots,n}_{m-1}, n(m-1)}.$$
We call this number the $(m,n)$th Catalan number $C(m,n)$. Clearly,
$C(2,n)$ is the ordinary $n$th Catalan number $C_n$.
\end{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{section}{Generalization}
In this section, we prove the our main theorem.
We show that the generalized Catalan numbers $C(m,n)$ are given by
$$\frac{1}{n(m-1)+1}{2n(m-1) \choose \underbrace{n,\ldots,n}_{m-1},n(m-1)}.$$
\begin{theorem}
Let $\mathbb{R}^m$ be the $m$-dimensional vector space. Consider
\begin{equation*}
\begin{array}{c}
R_1:(x_1,x_2,\ldots,x_m)\longrightarrow(x_1+1,x_2,\ldots,x_m),\\
R_2:(x_1,x_2,\ldots,x_m)\longrightarrow(x_1,x_2+1,\ldots,x_m),\\
\vdots \\
R_m:(x_1,x_2,\ldots,x_m)\longrightarrow(x_1,x_2,\ldots,x_m+1),
\end{array}
\end{equation*}
be $m$ kinds of moves in $\mathbb{R}^m$
(i.e., $R_i$ denotes the move one unit parallel to the $x_i$-axis in the
positive direction). Then
the number of paths from $\textbf{0}=(\underbrace{0,\ldots,0}_{m})$ to the point
$N:=\bigl( \underbrace{n,\ldots,n}_{m-1},(m-1)n \bigr)$ using the moves
$R_1,R_2,\ldots,R_m$ such that the path never rises above the hyperplane
$x_m=x_1+\cdots+x_{m-1}$ is
$$\frac{1}{n(m-1)+1}{2n(m-1)\choose \underbrace{n,\ldots,n}_{m-1},n(m-1)}.$$
\end{theorem}
\begin{proof}
We call a path from $\textbf{0}$ to $N$ of
$n$ $R_1$'s, $n$ $R_2$'s, $\ldots$, $n$ $R_{m-1}$'s, and $(m-1)n$ $R_m$'s
acceptable if the path never rises above the hyperplane $x_m=x_1+\cdots+x_{m-1}$
and unacceptable otherwise. Let $A_n^m$ and $U_n^m$ denote the number of
acceptable and unacceptable paths, respectively.
It is easy to see that each path from $\textbf{0}$
to $N$ corresponds to an arrangement
of $n$ $R_1$'s, $n$ $R_2$'s, $\ldots$, $n$ $R_{m-1}$'s, and $(m-1)n$ $R_m$'s.
Then
\begin{equation*}
A_n^m+U_n^m=\frac{\bigl( 2n(m-1)\bigr)!}{n!^{m-1}\bigl( n(m-1)\bigr)!}.
\end{equation*}
Now, consider an unacceptable path and its arrangement
$r_1,r_2,\ldots,r_{2n(m-1)}$, where $r_i\in\{R_1,R_2,\ldots,R_m\}$ indicates
the $i$th step of the path.
Since the path rises above the hyperplane, there is a first $t$
such that the number of $R_m$'s in $r_1,\ldots,r_t$ exceeds the sum of the
numbers $R_1,R_2,\ldots,R_{m-1}$. Moreover, $r_t=R_m$.
We only change $r_{t+1},\ldots,r_{2n(m-1)}$ the part of the path after the
crossing in the arrangement as follows:
Mark all the positions of the $R_m$'s in that part of the path and fill those
positions with the sequence (in order) consisting of all but the last of the
non-$R_m$'s. Then replace those non-$R_m$'s that have been used in the
replacement with $R_m$'s. Here is an example: let $m=3$, $n=2$ and the path
be given by $R_1 R_3 R_2 R_3 R_3 R_1 R_2 R_3$.
Then $t=5$, and the part of the path to be modified is $R_1 R_2 R_3$.
There is just one position of the $R_m$'s, so we replace $R_3$ with $R_1$,
and then fill the $R_1 R_2$ with $R_3 R_3$ to obtain
the modified sequence $R_1 R_3 R_2 R_3 R_3 R_3 R_3 R_1$.
The resulting arrangement
$r_1^{'},r_2^{'},\ldots,r_{2n(m-1)}^{'}$
is an arrangement of $(m-1)n+1$ $R_m$'s, $n$ $R_1$'s, $\ldots$, $n$ $R_{i-1}$'s,
$n$ $R_{i+1}$'s, $\ldots$, $n$ $R_{m-1}$'s, and $n-1$ $R_i$'s
for a $1\leq i\leq m-1$.
It is not difficult to see that this process is reversible:
\begin{equation*}
\begin{array}{c}
r_1,r_2,\ldots,r_{2n(m-1)}\\
\Updownarrow\\
\underbrace{*,*,\ldots,*,R_m}_{\substack{r R_m \text{'s, } \\ r-1 R_1,\ldots,R_{m-1}\text{'s.}}} \quad |%
\underbrace{*,*,\ldots,*}_{\substack{(m-1)n-r R_m\text{'s, } \\ (m-1)n-r+1 R_1,\ldots,R_{m-1}\text{'s.}}}\\
\Updownarrow\\
\underbrace{*,*,\ldots,*,R_m}_{\substack{r R_m\text{'s, } \\ r-1 R_1,\ldots,R_{m-1}\text{'s.}}} \quad |%
\underbrace{*,*,\ldots,*}_{\substack{(m-1)n-r+1 R_m\text{'s, } \\ (m-1)n-r R_1,\ldots,R_{m-1}\text{'s.}}}\\
\Updownarrow\\
r_1^{'},r_2^{'},\ldots,r_{2n(m-1)}^{'}
\end{array}
\end{equation*}
Hence, there are as many unacceptable arrangements as there are arrangements
of $(m-1)n+1$ $R_m$'s, $n$ $R_1$'s, $\ldots$, $n$ $R_{i-1}$'s, $n$ $R_{i+1}$'s,
$\ldots$, $n$ $R_{m-1}$'s, and $n-1$ $R_i$'s for a $1\leq i\leq m-1$. Then
\begin{equation*}
U_n^m=(m-1)\frac{\bigl(2n(m-1)\bigr)!}{n!^{m-2}(n-1)!\bigl(n(m-1)+1\bigr)!}.
\end{equation*}
So,
\begin{align*}
A_n^m=&\frac{\bigl(2n(m-1)\bigr)!}{n!^{m-1}\bigl(n(m-1)\bigr)!}-(m-1)%
\frac{\bigl(2n(m-1)\bigr)!}{n!^{m-2}(n-1)! \bigl(n(m-1)+1\bigr)!}\\
=&\frac{\bigl(2n(m-1)\bigr)!}{n!^{m-2}(n-1)!\bigl(n(m-1)\bigr)!}%
\Bigl(\frac{1}{n}-\frac{m-1}{n(m-1)+1}\Bigr)\\
=&\frac{\bigl(2n(m-1)\bigr)!}{n!^{m-1}\bigl(n(m-1)+1\bigr)!}\\
=&\frac{1}{n(m-1)+1}{2n(m-1) \choose \underbrace{n,\ldots,n}_{m-1},n(m-1)}.
\end{align*}
\end{proof}
We denote $A_n^m$ in the above proof by $C(m,n)$. The first few generalized Catalan
numbers are evaluated to be
\begin{center}
\[
\begin{array}{r|rrr}
n\backslash m & 3 & 4 & 5\\
\hline
0 & 1 & 1 & 1\\
1 & 4 & 30 & 336\\
2 & 84 & 11880 & 3603600\\
3 & 2640 & 8168160 & 76881235200\\
4 & 100100 & 7207615800 & 2229760743210000\\
5 & 4232592 & 7336632122820 & 77015151194691790080 \\
6 & 192203088 & 8193001579963200 & 2978057806800232994982144 \\
7 & 9178678080 & 9763825599821779200 & 124625746332992720112321024000 \\
8 & 455053212900 & 12209602888667136003480 & 5529032167369807343550830945418000
\end{array}
\]
\end{center}
\end{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{section}{Acknowledgments}
The author would like to thank the referee for his/her valuable comments and
suggestions which have improved the clarity of the proof of the Theorem 1.
This work is partially supported by the University of Kashan
under grant number 233437/3.
\end{section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{5}
\bibitem{B} R. A. Brualdi, {\it Introductory Combinatorics}, 5th ed.,
Prentice-Hall, 2009.
\bibitem{G} R. P. Grimaldi, {\it Discrete and Combinatorial Mathematics},
5th ed., Addison-Wesley, 2004.
\bibitem{Go} H. W. Gould, {\it Combinatorial Identities}, Morgantown,
West Virginia, 1972.
\bibitem{Go1} H. W. Gould, {\it Fundamentals of Series}, eight
tables based on seven unpublished manuscript notebooks (1945--1990),
edited and compiled by J. Quaintance, May 2010.
Available at \url{http://www.math.wvu.edu/~gould/}.
\bibitem{HLM} S. Heubach, N. Y. Li, and T. Mansour, Staircase tilings and
$k$-Catalan structures, {\it Discrete Math.} {\bf 308} (2008), 5954--5964.
\bibitem{HP} P. Hilton and J. Pedersen, Catalan numbers, their generalization,
and their uses, {\it Math. Intelligencer} {\bf 13} (2) (1991), 64--75.
\bibitem{K} D. Kim, On the $(n,k)$-th Catalan numbers,
{\it Commun. Korean Math. Soc.} {\bf 23} (2008), 349--356.
\bibitem{P} I. Pak, Reduced decompositions of permutations in terms of
star transpositions, generalized Catalan numbers and $k$-ary trees,
{\it Discrete Math.} {\bf 204} (1999), 329--335.
\bibitem{R} S. L. Richardson, Jr., {\it Enumeration of the generalized
Catalan numbers}, M.Sc. Thesis, Eberly College of Arts and Sciences,
West Virginia University, Morgantown, West Virginia, 2005.
\bibitem{S1} R. P. Stanley, {\it Enumerative Combinatorics}, Vol. 2,
Cambridge University Press, 1999.
\bibitem{S2} R. P. Stanley, {\it Catalan addendum}, preprint, May 25 2013.\newline
Available at \url{http://www-math.mit.edu/~rstan/ec/catadd.pdf}.
\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent 2000 {\it Mathematics Subject Classification}:
Primary 05A19; Secondary 05A10, 05A15.
\noindent \emph{Keywords:} Catalan number, path.
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequence \seqnum{A000108}.)
\bigskip
\hrule
\bigskip
\noindent
Received March 16 2013;
revised version received July 3 2013.
Published in {\it Journal of Integer Sequences}, July 30 2013.
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\noindent
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